sin(x+y)cos(x-y) = (sinx/secx) + (cosyy/cscy)

prove that both sides equal each other

On the left hand side you can apply the identities sin(x+y) = sin(x)cos(y)+cos(x)sin(y) and cos(x-y) = cos(x)cos(y)+sin(x)sin(y). On the right hand side replace the sec(x) and csc(y) terms by their sine and cosine equivalents. It comes right out after some algebra manipulation (and remember that cos^2a + sin^2a = 1).